## Geometry & Topology

### A very special EPW sextic and two IHS fourfolds

#### Abstract

We show that the Hilbert scheme of two points on the Vinberg $K3$ surface has a two-to-one map onto a very symmetric EPW sextic $Y$ in $ℙ5$. The fourfold $Y$ is singular along $60$ planes, $20$ of which form a complete family of incident planes. This solves a problem of Morin and O’Grady and establishes that $20$ is the maximal cardinality of such a family of planes. Next, we show that this Hilbert scheme is birationally isomorphic to the Kummer-type IHS fourfold $X0$ constructed by Donten-Bury and Wiśniewski [On 81 symplectic resolutions of a 4–dimensional quotient by a group of order $32$, preprint (2014)]. We find that $X0$ is also related to the Debarre–Varley abelian fourfold.

#### Article information

Source
Geom. Topol., Volume 21, Number 2 (2017), 1179-1230.

Dates
Revised: 28 January 2016
Accepted: 3 March 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510859177

Digital Object Identifier
doi:10.2140/gt.2017.21.1179

Mathematical Reviews number (MathSciNet)
MR3626600

Zentralblatt MATH identifier
1368.14016

#### Citation

Donten-Bury, Maria; van Geemen, Bert; Kapustka, Grzegorz; Kapustka, Michał; Wiśniewski, Jarosław. A very special EPW sextic and two IHS fourfolds. Geom. Topol. 21 (2017), no. 2, 1179--1230. doi:10.2140/gt.2017.21.1179. https://projecteuclid.org/euclid.gt/1510859177

#### References

• M F Atiyah, R Bott, A Lefschetz fixed point formula for elliptic complexes, II: Applications, Ann. of Math. 88 (1968) 451–491
• A Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983) 755–782
• G Bellamy, T Schedler, A new linear quotient of ${\bf C}^4$ admitting a symplectic resolution, Math. Z. 273 (2013) 753–769
• C Birkenhake, H Lange, Complex abelian varieties, 2nd edition, Grundl. Math. Wissen. 302, Springer (2004)
• W Bosma, J Cannon, C Playoust, The Magma algebra system, I: The user language, J. Symbolic Comput. 24 (1997) 235–265
• O Debarre, Annulation de thêtaconstantes sur les variétés abéliennes de dimension quatre, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987) 885–888
• L Dixon, J A Harvey, C Vafa, E Witten, Strings on orbifolds, Nuclear Phys. B 261 (1985) 678–686
• I V Dolgachev, Classical algebraic geometry: a modern view, Cambridge University Press (2012)
• I Dolgachev, Corrado Segre and nodal cubic threefolds, preprint (2015)
• I V Dolgachev, D Markushevich, Mutually intersecting planes in ${\mathbb P}^5$, Enriques surfaces, cubic $4$–folds, and EPW-septics, Oberwolfach Rep. 7 (2010) 1603–1605
• M Donten-Bury, J A Wiśniewski, On $81$ symplectic resolutions of a $4$–dimensional quotient by a group of order $32$ (2014) To appear in Kyoto J. Math.
• D Eisenbud, S Popescu, C Walter, Lagrangian subbundles and codimension $3$ subcanonical subschemes, Duke Math. J. 107 (2001) 427–467
• B van Geemen, J Top, An isogeny of $\mathrm{K3}$ surfaces, Bull. London Math. Soc. 38 (2006) 209–223
• D R Grayson, M E Stillman, Macaulay2, a software system for research in algebraic geometry Available at \setbox0\makeatletter\@url http://www.math.uiuc.edu/Macaulay2/ {\unhbox0
• P Griffiths, J Harris, Principles of algebraic geometry, Wiley-Interscience, New York (1978)
• G Hoehn, G Mason, Finite groups of symplectic automorphisms of hyperkähler manifolds of type $K3^{[2]}$, preprint (2014)
• G Kapustka, On IHS fourfolds with $b_2=23$, Michigan Math. J. 65 (2016) 3–33
• J Kollár, Shafarevich maps and plurigenera of algebraic varieties, Invent. Math. 113 (1993) 177–215
• H Maschke, Ueber die quaternäre, endliche, lineare Substitutionsgruppe der Borchardt'schen Moduln, Math. Ann. 30 (1887) 496–515
• U Morin, Sui sistemi di piani a due a due incidenti, Atti Istituto Veneto 89 (1930) 907–926
• K G O'Grady, Involutions and linear systems on holomorphic symplectic manifolds, Geom. Funct. Anal. 15 (2005) 1223–1274
• K G O'Grady, Irreducible symplectic $4$–folds and Eisenbud–Popescu–Walter sextics, Duke Math. J. 134 (2006) 99–137
• K O'Grady, Double covers of EPW-sextics, Michigan Math. J. 62 (2013) 143–184
• K G O'Grady, Pairwise incident planes and hyperkähler four-folds, from “A celebration of algebraic geometry” (B Hassett, J McKernan, J Starr, R Vakil, editors), Clay Math. Proc. 18, Amer. Math. Soc., Providence, RI (2013) 553–566
• Y G Prokhorov, Fields of invariants of finite linear groups, from “Cohomological and geometric approaches to rationality problems” (F Bogomolov, Y Tschinkel, editors), Progr. Math. 282, Birkhäuser, Boston (2010) 245–273
• R Varley, Weddle's surfaces, Humbert's curves, and a certain $4$–dimensional abelian variety, Amer. J. Math. 108 (1986) 931–951
• E B Vinberg, The two most algebraic $\mathrm{K3}$ surfaces, Math. Ann. 265 (1983) 1–21